1. Field of the Invention
Embodiments of the present invention relate generally to magnetic disk drives and, more particularly, to characterizing the frequency response of multirate systems, such as magnetic disk drives.
2. Description of the Related Art
In the vernacular of feedback control systems, a conventional hard disk drive (HDD) has a closed-loop servo control system comprising a “plant” with one or more moveable elements, e.g., a spindle motor, magnetic disk, and actuator assembly, and a servo controller for driving the moveable elements. Multirate controllers, which generate two or more corrections for each position measurement, are one type of servo controller commonly used in HDDs. Multirate controllers, also referred to as oversampling controllers, are necessary to position the read/write head of an HDD with respect to the center of a data track contained therein with the precision required by the high linear densities now in use for data storage on modern HDDs.
For the design and/or modeling of such controllers, it is advantageous to characterize the open-loop frequency response of the plant, i.e., the magnitude and phase of the mechanical response of the plant at each frequency of interest. This is because the need to understand the behavior of the system when excited at various frequencies appears in almost every practical control system problem, i.e., the open-loop frequency response is routinely considered in order to predict the system's closed-loop behavior. In the case of an HDD system, where the sampling frequency of the output is fixed by the design of the disk drive, characterizing the higher frequency dynamics and resonant frequencies of the system helps in the design or tuning of the servo controller parameters to improve the performance of the servo controller. Improved servo controller performance allows more precise tracking of the disk drive head with respect to the center of the data tracks, which in turn facilitates disk drives having greater storage capacity.
To characterize the frequency response of an HDD plant, the digitized output of the servo control system is typically used. Due to the nature of digital control systems, however, aliasing of the actual mechanical response of the plant occurs. That is, alias frequency components are generated as a result of digitally sampling a signal, i.e., the plant output, when the signal contains frequency components above the Nyquist frequency of the nominal sampling rate. The alias frequency components are measured as actual mechanical resonances, but in fact are only artifacts of the low sampling rate.
According to Shannon's sampling theorem, aliasing makes it impossible to recover the information lost during digital conversion if the sampling frequency is less than half the maximum frequency present in the signal. Aliasing can be minimized or avoided by increasing the sampling frequency well above the marginal sampling frequency dictated by Shannon's sampling theorem. In HDD systems, however, increasing the sampling frequency is generally impracticable, since the disk rotation speed and number of position data sectors on the disk are fixed. Thus, for signal reconstruction to be possible, the limiting frequency is half the sampling frequency, i.e. the Nyquist frequency, and the dynamics and resonant modes of an HDD system corresponding to frequencies close to and above half of the sampling frequency of the HDD cannot be identified with conventional methods.
The conventional method of quantifying the frequency response of a closed loop system is to excite the system with a sinusoidal sweep, record the sampled output data, and use a fast Fourier transform method to calculate the values of the frequency response at selected frequency points. As described above, this method breaks down at frequencies close to and above half of the sampling frequency due to aliasing. Specifically, alias components are measured along with actual mechanical resonances, both above and below the Nyquist frequency of the system. Because the frequency response measurements of an HDD plant take place in a closed-loop environment, both the mechanical resonances and the alias components are fed back through the closed-loop control system being measured, thereby confounding the alias components with the real mechanical resonances of the system and rendering the resultant characterization of the frequency response of the system inaccurate at best.
In light of the above, there is a need in the art for a method of computing the frequency response of a system beyond the Nyquist frequency of the nominal sample frequency.